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The words "microdifferential systems in the complex domain" refer to seve­ ral branches of mathematics: micro local analysis, linear partial differential equations, algebra, and complex analysis. The microlocal point of view first appeared in the study of propagation of singularities of differential equations, and is spreading now to other fields of mathematics such as algebraic geometry or algebraic topology. How­ ever it seems that many analysts neglect very elementary tools of algebra, which forces them to confine themselves to the study of a single equation or particular square matrices, or to carryon heavy and non-intrinsic formula­ tions when studying more general systems. On the other hand, many alge­ braists ignore everything about partial differential equations, such as for example the "Cauchy problem", although it is a very natural and geometri­ cal setting of "inverse image". Our aim will be to present to the analyst the algebraic methods which naturally appear in such problems, and to make available to the algebraist some topics from the theory of partial differential equations stressing its geometrical aspects. Keeping this goal in mind, one can only remain at an elementary level.




In this book we shall treat systems of differential or microdifferential equations, that is, modules (or sheaves of modules) over various rings of differential or microdifferential operators on a complex manifold.
Pierre Schapira

Chapter I. Microdifferential Operators

After recalling the principal notions concerning the Ring1) DX of holomorphic differential operators on a complex manifold X, we introduce the Ring \({\hat E_x}\) of formal microdifferential operators on the cotangent bundle T*X, assuming for a while that X is an open set in a complex (finite dimensional) vector space. This Ring is defined as a sheaf of formal series of homogeneous holomorphic functions, the composition law being obtained by extending the Leibniz formula. Then by means of the quasi-norms of L. Boutet de Monvel and P. Kree [1], we construct the sub-Ring ℰX of \({\hat E_x}\) of microdifferential operators of M. Sato, M. Kashiwara, T. Kawaï [1], and prove the invertibility of operators whose principal symbols do not vanish.
We construct various Banach algebras of zero order microdifferential operators, and use these algebras to obtain the extension to microdifferential operators of the classical division theorems of Spath and Weierstrass, as stated in Sato-Kashiwara-Kawaï [1], and also an extension of the Cauchy-Kowalewski theorem.
The division theorem asserts that if the principal symbol of an operator P has a zero exactly of order p in some direction \(\frac{\partial }{{\partial {\xi _1}}},\), then any operator Q may be divided by P with a remainder which is a polynomial of order at most p – 1 in \({D_{{x_1}}}\), having as coefficients microdifferential operators independent of \({D_{{x_1}}}\).
Roughly speaking the Cauchy-Kowalewski theorem asserts that if (x)=(x1,…,xn) are coordinates on X, the Cauchy Problem with values in the sheaf of operators which do not depend on \({D_{{x_1}}},...,{D_{{x_p}}},\), is well posed for operators P which are polynomials in \({D_{{x_1}}},...,{D_{{x_p}}},\), with zero order microdifferential operators as coefficients (and of course, with data on a non characteristic hypersurface). The proof is an application of the abstract Cauchy-Kowalewski theorem in scales of Banach spaces. We recall here this theorem with proof, following F. Trèves [1].
The left DX-module BZ|X, associated to a submanifold Z of X, and its microlocalisation CZ|X, are constructed in § 4. We avoid the cohomological tools, by associating a left Ideal of DX to a system of functions defining Z, and proving the intrinsic character of the constructions.
Then, following Sato-Kashiwara-Kawaï [1] and Kashiwara [5], we are ready to prove that given a complex contact transformation φ from an open set \(U \subset T*X\) to an open set \(U' \subset T*X',\phi \) can be locally “quantized”, that is extended to an isomorphism \(\hat \phi \) of filtered Rings from \({E_x}{|_u}\) to \({\phi ^{ - 1}}({E_{x'}}{|_{u'}})\). In fact if we set \(\Lambda _\phi ^a = \{ (x,x';\xi ,\xi ');(x'; - \xi ') = \phi (x,\xi )\} \), we have to find an Ideal ℐ of –X×X’, whose symbol Ideal coincides with the defining Ideal of \(\Lambda _\phi ^a\). Then we must prove, by successive applications of the division theorem, that given a section P of ℰX there exists a unique section Q of ℰX’ such that P—Q belongs to ℐ. When φ is the identity, ℐ is naturally associated to a volume element dx on X, and the anti-isomorphism P ↦ Q is nothing but the adjoint with respect to dx. In the general case \(\hat \phi \) is the composite of an antiisomorphism associated to \(\Lambda _\phi ^a\) and of the adjoint, for a volume element. We discuss some examples of quantized contact transformations, and in particular the action on ℰX of a complex change of coordinates. This allows us to define now ℰX when X is a complex manifold.
Once we are able to make use of quantized contact transformations, the theory of systems with “simple characteristics” becomes transparent, as shown in Sato-Kashiwara-Kawaï [1] Such a system ℳ is a left ℰX-module endowed with a generator u such that, if ℐ denotes the left Ideal of ℰX annihilating u, the symbol Ideal ℐ of ℐ coincides with JV, the defining Ideal of a smooth conic involutive manifold V. When V is regular involutive, V is isomorphic by a contact transformation to the manifold \(V' = \{ (x,\xi ) \in T*{\mathbb{C}^n};{\xi _1} = ... = {\xi _p} = 0,{\xi _n} \ne 0\} \), and successive applications of the Cauchy-Kowalewski theorem permit us to prove that ℳ is then locally isomorphic to a module ℰX/J, where J is the left Ideal generated by \({D_{{x_1}}},...,{D_{{x_p}}}\). When V is Lagrangean, it is isomorphic to \(V' = \{ (x,\xi ) \in T*{\mathbb{C}^n};{x_1} = {\xi _2} = ... = {\xi _n} = 0,{\xi _1} \ne 0\} \), the conormal bundle to the hypersurface {x1=0{, and there exists a complex number α, unique modulo ℤ, such that ℳ=ℰX/ J, the Ideal J being generated by \({x_1}{D_{{x_1}}} - a,{D_{{x_2}}},...,{D_{{x_n}}}\).
The notions of symplectic geometry used in this Chapter are recalled in Appendix A.
Pierre Schapira

Chapter II. ℰ X -modules

This chapter is devoted to the study of the algebraic structure of the sheaf of rings ℰX and of the usual operations on Modules over ℰX. The Ring ℰX has a Z-filtration induced by the order, and many properties of a ℤ-filtered Module M over a ℤ-filtered Ring A can be obtained from the corresponding ones on gr(M), the associated graded Module over gr(A), if we make the assumption that the filtration is what we call a “zariskian filtration”. We study the theory of filtered modules in the abstract in § 1, with special attention to characteristic ideals, flatness, homological dimension, coherency. The important theorem of O. Gabber [1] on the involutivity of the characteristic ideal is stated, but it seemed irrelevant to us to duplicate his very clear proof.
Next we have to show that the filtration over ℰX is in fact a zariskian one. The proof follows that of Sato-Kashiwara-Kawaï [1] and is classical in its approach: induction on the dimension with the help of the division theorems. The main properties of ℰX then follow immediately: the sheaf ℰX is coherent and noetherian, and the characteristic cycle of a coherent ℰX-module is a conic analytic involutive cycle. As a corollary of the involutivity we get that the homological dimension of ℰX is equal to n the (complex) dimension of X, and even that any coherent Module locally admits a free resolution of length at most n.
It is beyond the scope of this book to make a detailed study of holonomic Modules (i.e.: coherent Modules whose characteristic variety is Lagrangean). We just give the first properties of the category of holonomic Modules, and study the “adjoint” functor on this category.
We end § 2 by introducing the “dummy variable” trick which enables to work outside of the zero section of T*X, and by finally studying the relations between coherent DX-modules and globally defined coherent ℰX-modules.
Next we explain how one manipulates coherent gℰX-modules with the same ease as is possible for functions: tensor products for two Modules over two different manifolds, inverse images for left ℰX-modules (analogous to restriction for functions), direct images of right Modules (in line with integration of differential forms). All these constructions have been done in Sato-Kashiwara-Kawaï [1] using a deep cohomological method, but we proceed here in an elementary way, beginning with the study of these operations for Modules like IX|X, then passing to the general case. Non-characteristic restrictions of ℰX-modules are studied in detail, and we calculate exactly, following M. Kashiwara [5], the characteristic cycle of the induced system in this case.
On the other hand direct images for proper maps are not studied here, and reference is made to Kashiwara [3] and Pham [1] for further developments on this subject (cf. also Chapter III, § 4).
Finally we give a general formulation of the Cauchy problem for general systems, as preparation for the next chapter.
Pierre Schapira

Chapter III. Cauchy Problem and Propagation

In many cases, the study of the holomorphic solutions of systems of differential equations, that is, of the sheaves \(Ext_{{D_x}}^j(M,{O_x})\), reduces to the study of char(ℳ), the characteristic variety of ℳ.
Pierre Schapira


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